# Canonical Quantization of Noncompact Spin System

## TL;DR

This paper develops a method for quantizing noncompact spin systems with SU(N,1) symmetry by reducing the classical system to a symplectic manifold and solving the Schrödinger equation exactly.

## Contribution

It provides a direct constraint solution for noncompact spins and demonstrates canonical quantization on the reduced phase space with an exact propagator.

## Key findings

- Successfully formulated noncompact spins as functions of canonical variables
- Achieved canonical quantization on the reduced phase space
- Derived the exact propagator for the quantum system

## Abstract

We consider spin system defined on the coadjoint orbit with noncompact symmetry and investigate the quantization. Classical spin with noncompact SU(N,1) symmetry is first formulated as a dynamical system and the constraint analysis is performed to reduce the system from the group space to the coadjoint orbit which is a symplectic manifold with Kahler structure. We achieve this by solving the constraint directly. It is shown that the dynamical variables describing the noncompact spins can be written as functions of canonically conjugate variables and canonical quantization is possible on the reduced phase space. With the quantum mechanical Hamiltonian acting on the holomorphic coherent state in Hilbert space, we obtain the exact propagator by solving the time-dependent Schrodinger equation.